Quantum quench in p+ip superfluids: Winding numbers and topological - PowerPoint PPT Presentation

Quantum quench in p+ip superfluids: Winding numbers and topological states far from equilibrium Matthew S. Foster, 1 Maxim Dzero, 2 Victor Gurarie, 3 and Emil A. Yuzbashyan 4 1 Rice University, 2 Kent State University, 3 University of Colorado at Boulder, 4 Rutgers University April 26 th , 2013

What is a “quantum quench”? A non-adiabatic perturbation to a closed quantum many-particle system

Quantum Quench: Coherent many-body evolution Quantum quench protocol 1. Prepare initial state e.g. • : ground state of • Assume excitation gap

Quantum Quench: Coherent many-body evolution Quantum quench protocol 1. Prepare initial state 2. “Quench” the Hamiltonian: Non-adiabatic perturbation

Quantum Quench: Coherent many-body evolution Quantum quench protocol 1. Prepare initial state 2. “Quench” the Hamiltonian: Non-adiabatic perturbation 3. Exotic excited state, coherent evolution

Quantum Quench: Coherent many-body evolution  Decoherence, dissipation…

Quantum Quench: Coherent many-body evolution New fields of non-equilibrium dynamics 1. Nanostructures, qubits 2. Ultrafast spectroscopy 3. Ultracold atoms Extreme isolation → Long relaxation times • • Highly tunable Experimental Example: Collapse and revival of matter wave interference: SF to Mott quench in a boson atom optical lattice Bloch, Dalibard, Zwerger 2008 Greiner, Mandel, Hänsch, and Bloch 2002

Quantum Quench: Coherent many-body evolution Experimental Example: Quantum Newton’s Cradle for trapped 1D Bose Gas Kinoshita, Wenger, and Weiss 2006

The theoretical story so far: Quantum quenches  Thermalization Rigol, Dunjko, Olshannii (2008); Rossini, Silva, Mussardo, Santoro (2009)...  Quantum critical scaling Polkovnikov (2005), De Grandi, Gritsev, Polkovnikov (2010) …  Post-quench quasiparticle distribution functions Rigol, Dunjko, Yurovsky, Olshannii (2007) …  Post-quench evolution of correlation functions Calabrese and Cardy (2006); Gritsev, Demler, Lukin, Polkovnikov (2007) ...  Transverse field Ising Cincio, Dziarmaga, Rams, Zurek 2007; Rossini, Silva, Mussardo, Santoro 2009 Calabrese, Essler, Fagotti 2011; Schuricht and Essler 2012; Calabrese, Essler, and Fagotti 2012  Lieb-Liniger Demler, Imambekov, Kormos, Iyer, Andrei, Gritsev, Mossel, Caux, Buljan, Pezer, Gasenzer, Konik...  “Quantum” solitons Foster, Altshuler, and Yuzbashyan (2010), Foster, Berkelbach, Reichman, and Yuzbashyan (2011); Neuenhahn, Polkovnikov, Marquardt (2012)

P-wave superconductivity in 2D Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian

P-wave superconductivity in 2D Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian Anderson pseudospins

P-wave superconductivity in 2D Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian Anderson pseudospins Time-reversal particle pair { k ,- k } vacant No pair { k ,- k } vacant

P-wave superconductivity in 2D Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian Anderson pseudospins Fermi liquid state: Discontinuous domain wall

P-wave superconductivity in 2D Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian Anderson pseudospins Superconducting state: Smooth domain wall P. Anderson 1958

P-wave superconductivity in 2D Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian “P + i p” superconducting state: Skyrmion pseudospin texture G. E. Volovik 1988; Read and Green 2000

P-wave superconductivity in 2D Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian “P + i p” superconducting state: solidforms.com Fully gapped, non s-wave

P-wave superconductivity in 2D Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian “P + i p” superconducting state: At fixed density n: • µ is a monotonically decreasing function of ∆ 0 BCS BEC

Topological superconductivity in 2D Pseudospin winding number Q : BCS G. E. Volovik 1988; Read and Green 2000 2D Topological superconductor Fully gapped when µ ≠ 0 • BEC • Weak-pairing BCS state topologically non-trivial • Strong-pairing BEC state topologically trivial

Topological superconductivity in 2D Pseudospin winding number Q : G. E. Volovik 1988; Read and Green 2000 Retarded GF winding number W (i.e., compute G in BdG MFT): Niu, Thouless, and Wu 1985 G. E. Volovik 1988 • W = Q in ground state W ≠ 0 signals presence of chiral edge states •

Topological superconductivity in 2D Topological signatures: Majorana fermions 1. Chiral 1D Majorana edge states quantized thermal Hall conductance J. Moore 2. Isolated Majorana zero modes in type II vortices Realizations? 3 He-A thin films, Sr 2 RuO 4 (?) • Volovik 1988, Rice and Sigrist 1995 • 5/2 FQHE: Composite fermion Pfaffian Moore and Read 1991, Read and Green 2000 • Cold atoms Gurarie, Radzihovsky, Andreev 2005; Gurarie and Radzihovsky 2007 Zhang, Tewari, Lutchyn, Das Sarma 2008; Sato, Takahashi, Fujimoto 2009; Y. Nisida 2009 • Polar molecules Cooper and Shlyapnikov 2009; Levinsen, Cooper, and Shlyapnikov 2011 • S-wave proximity-induced SC on surface of 3D Z2 Top. Insulator Fu and Kane 2008

Topological protection 2D weak-pairing BCS p+ip superconductor: Fully-gapped, “strong” topological state (class D) Robust against • Weak (enough) disorder • Weak interaction perturbations (e.g., pair-breaking terms) Stability of topological order against 1. Strong disorder? 2. Hard non-equilibrium driving?

Topological protection … ? 2D weak-pairing BCS p+ip superconductor: Fully-gapped, “strong” topological state (class D) Robust against • Weak (enough) disorder • Weak interaction perturbations (e.g., pair-breaking terms) Stability of topological order against 1. Strong disorder? 2. Hard non-equilibrium driving? Topological Order vs. Quantum Quench ( Fight! ) • Entanglement entropy survival in the toric code following quench Rahmani and Chamon 2010 • Kibble-Zurek excitation probability in adiabatic p+ip chain quench DeGottardi, Sen, and Vishveshwara 2011

P-wave superconductivity in 2D: Dynamics 2D P-wave BCS Hamiltonian…“hard”

P-wave superconductivity in 2D: Dynamics Ibañez, Links, Sierra, and Zhao (2009): Chiral 2D P-wave BCS Hamiltonian • Same p+ip ground state, non-trivial (trivial) BCS (BEC) phase Richardson (2002) • Integrable (hyperbolic Richardson model) Dunning, Ibanez, Links, Sierra, and Zhao (2010) Rombouts, Dukelsky, and Ortiz (2010) • Reduces to 1D: Claim: for p+ip initial state, self-consistent mean-field dynamics are identical to “real” p-wave Hamiltonian

P-wave Quantum Quench • Initial p+ip BCS or BEC state: • Post-quench Hamiltonian: • Quench parameter: BCS BEC

P-wave Quantum Quench • Initial p+ip BCS or BEC state: • Post-quench Hamiltonian: • Quench parameter: BCS BEC

Chiral P-wave BCS: Dynamics Heisenberg spin equations of motion: Self-consistent mean field theory (thermodynamic limit) Identical to self-consistent (time-dependent) Bogoliubov-de Gennes

Chiral P-wave BCS: Dynamics Classical spin equations of motion:

Chiral P-wave BCS: Dynamics Classical spin equations of motion: Ground state: spins aligned with field!

Chiral P-wave BCS: Dynamics Classical spin equations of motion: Ground state: 1. “Gap equation” 2. Chemical potential vs density (Absorb linear divergence into G at QCP)

Chiral P-wave BCS: Lax construction Lax vector components, “norm” Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Chiral P-wave BCS: Lax construction Lax vector components, “norm” Generalized Gaudin algebra M. Gaudin 1972, 1976, 1983 Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Chiral P-wave BCS: Lax construction Lax vector components, “norm” Lax vector norm: Generates integrals of motion From Gaudin algebra: BCS Hamiltonian:

Chiral P-wave BCS: Lax construction Lax vector components, “norm” Yuzbashyan, Altshuler, Conserved spectral polynomial: Kuznetsov, and Enolskii (2005) Key to understanding ground state and quench dynamics

Ground state: Lax roots In the ground state (zero quench):

Ground state: Lax roots In the ground state (zero quench):

Ground state: Lax roots In the ground state (zero quench): One isolated pair of roots ; ( N – 1) positive, real, doubly-degenerate roots Gap, chemical potential encoded in isolated roots

P-wave quantum quench: Lax roots • Initial p+ip BCS or BEC state: • Post-quench Hamiltonian: • Quench parameter: Roots: Strong-to-weak quench #1 ( β > 0) Quench

P-wave quantum quench: Lax roots • Initial p+ip BCS or BEC state: • Post-quench Hamiltonian: • Quench parameter: Roots: Strong-to-weak quench #1 ( β > 0) Quench 1 isolated pair 100 spins (numerics)

P-wave quantum quench: Lax roots • Initial p+ip BCS or BEC state: • Post-quench Hamiltonian: • Quench parameter: Roots: Strong-to-weak quench #2 ( β > 0) Quench No isolated pair 100 spins (numerics)

P-wave quantum quench: Lax roots • Initial p+ip BCS or BEC state: • Post-quench Hamiltonian: • Quench parameter: Roots: Weak-to-strong quench #1 ( β < 0) Quench 2 isolated pairs 100 spins (numerics)